Theorem xmstopn 24576

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
xmstopn	⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 24572	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	4	simprbi 502	1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   × cxp 5660   ↾ cres 5664  ‘cfv 6537  Basecbs 17268  distcds 17318  TopOpenctopn 17473  MetOpencmopn 21480  TopSpctps 23057  ∞MetSpcxms 24442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-res 5674  df-iota 6493  df-fv 6545  df-xms 24445

This theorem is referenced by:  imasf1oxms  24614  ressxms  24650  prdsxmslem2  24654  tmsxpsmopn  24662  xmsusp  24694  cmetcusp1  25480  minveclem4a  25557  minveclem4  25559  qqhcn  34325  rrhcn  34331  rrexthaus  34341  dya2icoseg2  34612  sitmcl  34685